A326029 Number of strict integer partitions of n whose mean and geometric mean are both integers.

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 1, 2, 1, 3, 1, 1, 2, 3, 1, 3, 1, 1, 3, 6, 1, 3, 1, 2, 1, 1, 1, 3, 1, 6, 1, 5, 1, 2, 2, 2, 4, 3, 1, 9, 1, 1, 3, 1, 1, 4, 1, 4, 2, 6, 1, 6, 1, 3, 7, 4, 2, 5, 1, 10, 1, 3, 1, 9, 3

Offset

0,11

Links

Table of n, a(n) for n=0..85.

Wikipedia, Geometric mean

Example

The a(55) = 2 through a(60) = 9 partitions:
  (55)           (56)         (57)        (58)    (59)  (60)
  (27,16,9,2,1)  (24,18,8,6)  (49,7,1)    (49,9)        (54,6)
                              (27,25,5)   (50,8)        (48,12)
                              (27,18,12)                (27,24,9)
                                                        (27,24,6,2,1)
                                                        (36,12,9,2,1)
                                                        (36,9,6,4,3,2)
                                                        (24,18,9,6,2,1)
                                                        (27,16,9,4,3,1)

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&IntegerQ[Mean[#]]&&IntegerQ[GeometricMean[#]]&]], {n, 0, 30}]

Crossrefs

Partitions with integer mean and geometric mean are A326641.

Strict partitions with integer mean are A102627.

Strict partitions with integer geometric mean are A326625.

Non-constant partitions with integer mean and geometric mean are A326641.

Subsets with integer mean and geometric mean are A326643.

Heinz numbers of partitions with integer mean and geometric mean are A326645.

Cf. A051293, A067538, A067539, A078175, A082553, A316413, A326027, A326623, A326644, A326646, A326647.

Sequence in context: A074298 A356097 A356096 * A356167 A176028 A375391

Adjacent sequences: A326026 A326027 A326028 * A326030 A326031 A326032

Keyword

nonn

Author

Gus Wiseman, Jul 16 2019

Extensions

More terms from Jinyuan Wang, Jun 26 2020

Status

approved